Denominator identities and Lie superalgebras Paolo Papi Sapienza - PowerPoint PPT Presentation

Denominator identities and Lie superalgebras Paolo Papi Sapienza Universit` a di Roma We find an analogue of the Weyl denominator identity for a basic classical Lie superalgebra. joint work with Victor Kac and Pierluigi M¨ oseneder Frajria

Weyl denominator identity g complex finite-dimensional Lie algebra h Cartan subalgebra ∆ root system of ( g , h ) W Weyl group of ∆ ∆ + ⊂ ∆ set of positive roots, ρ = 1 � α ∈ ∆ + α 2 Theorem � � (1 − e − α ) = sgn ( w ) e w ( ρ ) − ρ α ∈ ∆ + w ∈ W For g = sl ( n ), Weyl formula is related to the expansion of the Vandermonde determinant.

Basic classical Lie superalgebras First we want to locate a suitable class of Lie superalgebras to which trying to extend the Weyl formula. These are the Basic classical Lie superalgebras g = g 0 ⊕ g 1 : • g is simple; • g 0 is a reductive Lie algebra; • g has a nondegenerate bilinear invariant form which is symmetric on g 0 , symplectic on g 1 and such that ( g 0 , g 1 ) = 0. h ⊂ g 0 Cartan subalgebra g = � g α weight space decomposition α ∈ ∆ ∆ = ∆ 0 ∪ ∆ 1 decomposition into even and odd “roots” W Weyl group of ∆ 0 .

Example: gl ( m , n ). �� A � A � A � 0 �� � � � B B 0 B gl ( m , n ) = , = + 0 0 C D C D D C � �� � � �� � degree 0 degree 1 is a Lie superalgebra w.r.t. [ X , Y ] = XY − ( − 1) deg ( X ) deg ( Y ) YX . m + n h = diagonal matrices , h ∗ = � C ε i , δ j = ε m + j , 1 ≤ j ≤ n . i =1 � A � B ( X , Y ) = str ( XY ) , str ( ) = tr ( A ) − tr ( D ) . C D The above bilinear form can be normalized in such a way that ( ε i , ε j ) = δ ij , ( δ h , δ k ) = − δ hk . ∆ 0 = ±{ ε i − ε j | 1 ≤ i < j ≤ m } ∪ ±{ δ i − δ j | 1 ≤ i < j ≤ n } ∆ 1 = ±{ ε i − δ j | 1 ≤ i ≤ m , 1 ≤ j ≤ n }

Why Weyl formula doesn’t work for superalgebras.... Because: 1. the restriction of ( · , · ) to Span R ∆ is usually indefinite; 2. the sets of positive roots are no more W -conjugate. Corresponding comments: 1. One defines the defect d of g as the dimension of a maximal isotropic subspace of � R α . It can be shown that d equals α ∈ ∆ the cardinality of a maximal isotropic subset of ∆ + (a subset S ⊂ ∆ + is isotropic if it is formed by linearly independent pairwise orthogonal isotropic roots); 2. it won’t be a surprize that formulas do depend on the choice of ∆ + .

Kac-Wakimoto-Gorelik Theorem Fix ∆ + . Define the Weyl-Kac denominator and superdenominator � � 0 (1 − e − α ) 0 (1 − e − α ) α ∈ ∆ + α ∈ ∆ + R + = 1 (1 + e − α ) , R − = � � 1 (1 − e − α ) α ∈ ∆ + α ∈ ∆ + � where ρ = ρ 0 − ρ 1 , ρ i = 1 i α, i = 0 , 1 . α ∈ ∆ + 2 Theorem Let ∆ + be any set of positive roots such that a maximal isotropic subset S of ∆ + is contained in the set of simple roots Π corresponding to ∆ + . Then e ρ � e ρ R ± = sgn ± ( w ) w β ∈ S (1 ± e − β ) . � w ∈ W ♯ where W # is a subgroup of W .

Main result Definition A set of positive root ∆ + is called distinguished if the corresponding set of simple roots has exactly one odd root. The following result is proven in the extended abstract, by representation theoretic methods. Theorem Let g = g 0 ⊕ g 1 be a basic classical Lie superalgebra of defect d, where g = A ( d − 1 , d − 1) is replaced by gl ( d , d ) . Then, for any distinguished set of positive roots, we have e ρ C e ρ R ± = X sgn ( w ) w (1 ± e − γ 1 )(1 − e − γ 1 − γ 2 ) · · · (1 + ± ( − 1) d +1 e − γ 1 − γ 2 − ... − γ d ) w ∈ W where { γ 1 , . . . , γ d } is an explicitly defined maximal isotropic subset of ∆ + and C is an explicit constant.

Comments • Distinguished sets of positive root exist for any g and have been classified by Kac. • For a distinguished ∆ + the hypothesis of KWG Theorem holds iff d = 1. This is a strong constraint, since, e.g. def gl ( m , n ) = min { m , n } . • Distinguished sets of positive root are quite often used when presenting g as a contragredient Lie superalgebra (roughly speaking: by generators and relations). • As we shall see, the proof follows quite naturally by Howe duality (and for type A , in a more elementary way, by Cauchy formulas).

The metaplectic representation The choice of a set of positive roots ∆ + determines a polarization � g 1 = g + 1 + g − 1 , where g ± 1 = g α . α ∈ ∆ ± 1 Hence we can consider the Weyl algebra W ( g 1 ) = T ( g 1 ) / � x ⊗ y − y ⊗ x − ( x , y ) � of ( g 1 , ( , ) | g 1 ) and construct the left W ( g 1 )-module M ∆ + ( g 1 ) = W ( g 1 ) / W ( g 1 ) g + 1 , The module M ∆ + ( g 1 ) is also a sp ( g 1 , ( , ))–module with T ∈ sp ( g 1 , ( , )) acting by left multiplication by � dim g 1 θ ( T ) = − 1 T ( x i ) x i , where { x i } is any basis of g 1 and 2 i =1 { x i } is its dual basis w.r.t. ( , ). Since ad ( g 0 ) ⊂ sp ( g 1 , ( , )), we obtain an action of g 0 on M ∆ + ( g 1 ).

The metaplectic representation We have a h -module isomorphism M ∆ + ( g 1 ) ∼ = S ( g − 1 ) ⊗ C − ρ 1 where S ( g − 1 ) is the symmetric algebra of g − 1 . Its h -character is e − ρ 1 chM ∆ + ( g 1 ) = 1 (1 − e − α ) . (1) � α ∈ ∆ + Upon multiplication by e ρ 0 � 0 (1 − e − α ) the r.h.s. of (1) α ∈ ∆ + becomes e ρ R − So if we are able to determine the g 0 -character of M ∆ + ( g 1 ) we have a formula for R − .

Example: sl ( m , n ) and Cauchy formulas In this case there is essentially a unique distinguished set of positive � 0 � B roots: g + 1 = . The special feature of this example is 0 0 that g − 1 is a g 0 -module , and the action of g 0 on g − 1 is the natural action of { ( A , B ) ∈ gl ( m + 1) × gl ( n + 1) | tr ( A ) + tr ( B ) = 0 } on ( C m +1 ) ∗ ⊗ C n +1 . Assume m > n . Cauchy formulas give � 1 )) = ch ( S (( C m +1 ) ∗ ⊗ C n +1 )) = ch ( S ( g − L A m ( τ ( λ )) L A n ( λ ) λ where for λ 1 ≥ λ 2 ≥ . . . ≥ λ n +1 we have set n +1 n +1 � � λ = λ i δ i , τ ( λ ) = − w 0 ( λ i ε i ) i =1 i =1 and w 0 is the longest element in the symmetric group W ( A m ).

Then � S ( g − L A m × A n ( − ρ 1 − s 1 γ 1 − . . . − s n +1 γ n +1 ) , 1 ) ⊗ C − ρ 1 = s 1 ≥ s 2 ≥ ... ≥ s n +1 γ 1 = ε m +1 − δ 1 , γ 2 = ε m − δ 2 , . . . . . . , γ n +1 = ε m − n +1 − δ n +1 . By the Weyl character formula, we have for s = s 1 ≥ s 2 ≥ . . . ≥ s n +1 e w ( λ s 1 ,..., sn +1 + ρ 0) e − ρ 1 chL Am × An ( λ s 1 ,..., sn +1 ) = X X X = sgn ( w ) . e ρ 0 Q (1 − e − β ) Q (1 − e − α ) s s w ∈ W β ∈ ∆+ β ∈ ∆+ 1 0 Hence � � e ρ R − = sgn ( w ) e w ( ρ − s 1 γ 1 − ... − s n +1 γ n +1 ) s 1 ≥ s 2 ≥ ... ≥ s n +1 w ∈ W � � sgn ( w ) e w ( ρ − s 1 γ 1 − s 2 ( γ 1 + γ 2 ) ... − s n +1 ( γ 1 + ... + γ n +1 )) s 1 , s 2 ,..., s n +1 w ∈ W

General case: Howe duality For a distinguished set of positive roots, we build up a real form V of g 1 endowed with a standard symplectic basis { e α , f α } α ∈ ∆ + 1 such 1 C ( e α ± √− 1 f α ) = g + that � 1 . It turns out that α ∈ ∆ + sp ( V ) ∩ ad ( g 0 ) = s 1 × s 2 , s i , i = 1 , 2 , being the Lie algebras of a compact dual pair ( G 1 , G 2 ) in Sp ( V ). Distinguished sets of positive roots turn out to correspond in this way to compact dual pairs ( G 1 , G 2 ), with G 1 compact: ∆ + ∆ + B → ( O (2 m + 1) , Sp (2 n , R )) , A → ( U ( m ) , U ( n )) , ∆ + ∆ + D 2 → ( Sp ( m ) , SO ∗ (2 n )) . D 1 → ( O (2 m ) , Sp (2 n , R )) , If λ → τ ( λ ) is the Theta correspondence, then, as g 0 -modules � ch M ∆ + ( g 1 ) = L G 1 ( λ ) L ( s 2 ) C ( τ ( λ )) . λ

Updating the main result.... By now we have proven WITH COMBINATORIAL METHODS the following theorem. Theorem Let g = g 0 ⊕ g 1 be a basic classical Lie superalgebra of defect d, where g = A ( d − 1 , d − 1) is replaced by gl ( d , d ) . Let ∆ + be any set of positive roots and Q + be the corresponding positive root lattice. There is a combinatorial procedure that starting from ∆ + yields a class of maximal isotropic sets in ∆ + . Fix any of them, say S special = { γ 1 , . . . , γ d } . Then we have e ρ � C · e ρ R ± = sgn ± ( w ) w � d i =1 (1 ± χ ± i e −� γ i � ) w ∈ W g for an explicit constant C. Moreover, there exists a choice for S special such that � γ i � ∈ Q + for any i = 1 , . . . , d .

Comments Undefined notation: � 1 if η ∈ Z ∆ 0 , ε ( η ) = − 1 if η ∈ Z ∆ \ Z ∆ 0 , γ ≤ i = { β ∈ S special , β ≤ γ i } , � � γ i � = ε ( γ i − β ) β, β ∈ γ ≤ i i = ( − 1) | γ ≤ χ + i | +1 , χ − i = 1 where β ≤ γ if γ − β is a sum of positive roots or zero. A nice feature of the theorem is that it allows us to recover the Theta correspondence for compact dual pairs starting from the combinatorial formula. This is highly non trivial, but it works!